Efficient Fast Marching with Finsler metrics.
Résumé
We study the discretization of the Escape Time problem: find the length of the shortest path joining an arbitrary point of a domain, to the domain's boundary. Path length is measured locally via a Finsler metric, potentially asymmetric and strongly anisotropic. This Optimal Control problem can be reformulated as a static Hamilton Jacobi, or Anisotropic Eikonal, Partial Differential Equation, as well as a front propagation model. It has numerous applications, ranging from motion planning to image segmentation. We introduce a new algorithm, Fast Marching using Anisotropic Stencil Refinement (FM-ASR), which addresses this problem on a two dimensional domain discretized on a cartesian grid. The local stencils used in our discretization are produced by arithmetic methods. The complexity of the FM-ASR, in an average sense over all grid orientations, only depends (poly-)logarithmically on the anisotropy ratio of the metric, while most alternative approaches have a polynomial dependence. Numerical experiments show, in several occasions, that the accuracy/complexity compromise is improved by an order of magnitude or more.
Domaines
Analyse numérique [math.NA]Origine | Fichiers produits par l'(les) auteur(s) |
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